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The standard paradigm for online power of two choices problems in random graphs is the Achlioptas process. Here we consider the following natural generalization: Starting with G 0 as the empty graph on n vertices, in every step a set of r edges is drawn uniformly at random from all edges that have not been drawn in previous steps. From these, one edge has(More)
Consider the following one-player game. Starting with the empty graph on n vertices, in every step r new edges are drawn uniformly at random and inserted into the current graph. These edges have to be colored immediately with r available colors, subject to the restriction that each color is used for exactly one of these edges. The player's goal is to avoid(More)
We study non-overlapping axis-parallel packings of 3D boxes with profits into a dedicated bigger box where rotation is either forbidden or permitted, and we wish to maximize the total profit. Since this optimization problem is NP-hard, we focus on approximation algorithms. We obtain fast and simple algorithms for the non-rotational scenario with(More)
The evolution of the largest component has been studied intensely in a variety of random graph processes, starting in 1960 with the Erd˝ os-Rényi process (ER). It is well known that this process undergoes a phase transition at n/2 edges when, asymptotically almost surely, a linear-sized component appears. Moreover, this phase transition is continuous, i.e.,(More)
The standard randomization of Ramsey's theorem [11] asks for a fixed graph F and a fixed number r of colors: for what densities p = p(n) can we asymptotically almost surely color the edges of the random graph G(n, p) with r colors without creating a monochromatic copy of F. This question was solved in full generality by Rödl and Ruci´nski [12, 14]. In this(More)
We study the following two problems: i) Given a random graph G n,m (a graph drawn uniformly at random from all graphs on n vertices with exactly m edges), can we color its edges with r colors such that no color class contains a component of size Θ(n)? ii) Given a random graph G n,m with a random partition of its edge set into sets of size r, can we color(More)
The internal diffusion limited aggregation (IDLA) process places n particles on the two dimensional integer grid. The first particle is placed on the origin; every subsequent particle starts at the origin and performs an unbiased random walk until it reaches an unoccupied position. In this work we study the computational complexity of determining the subset(More)